I have a question, and I am guessing that the question arises due to my lack of good understanding in the change of variable technique.
I would like to evaluate $f_X(x)$. When $f_Y(y)$ exists, I can write $f_X(x)$ as: $$ f_X(x) = \int f_{X|Y}(x|y) f_Y(y) dy $$ In the scenario that I encountered, $f_Y(y)$ is a multivariate normal with mean $\mu$ and variance $V$. Then $Y=\mu + A^T Z$ where $Z$ is a random variable following a standard multivariate normal distribution, and $A^T A =V$.
For some reasons, I would like to evaluate $f_X(x)$ in the way using $Z$ rather than $Y$. That is, I would like to evaluate:
$$ f_X(x) = \int f_{X|Z}(x|z) f(z) dz $$
In this case, what is the form of $f(x|z)$ ?
P.S.
I tried to solve this problem in this way
Since $Z= A^{-T}[Y-\mu]$, using the change of variable technique, we have $f_Y(y)=f_Z(A^{-T}[y-\mu])|A^{-1}|$. Therefore, I should be able to write $f(x)$ as:
$$ f_X(x) = \int f_{X|Y}(x|y) f_Z(A^{-T}[y-\mu])|A^{-1}|dy $$
However, this calculation did not take me anywhere....